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By Boelviken E.

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3 Acceptance–rejection Acceptance–rejection is a random stopping rule and much more subtle than inversion. The idea is to sample from a density function g(x) of our choice. Simulations that do not meet a certain acceptance criterion A are discarded, and the rest will then come from the original density function f (x). Magic? It works like this. Let g(x|A) be the density function of the simulations kept. , the probability that X = x drawn from g(x) is allowed to stand. 13) M≥ g(x) and suppose X is accepted whenever a uniform random number U satisfies U≤ f (x) .

1) i=1 with the statistical properties for the mean being well known: σ ∗ ∗ E(X − ξ) = 0 and sd(X ) = √ . 2) Monte Carlo estimates of ξ are unbiased, and their error may in theory be pushed √ ∗ below any level by raising m. 2) has been replaced by s∗ . This kind of uncertainty is often minor compared ∗ with other sources of error (see Chapter 7), but if X is used as the price of an asset or a transaction, high Monte Carlo accuracy may be necessary. 1 if kurtosis is unknown ground). For normal variables, κ = 0.

5. The latter is a forerunner of the entire Part I, where the Monte Carlo method and its potential are presented jointly with elementary models in insurance and finance. The idea is to get Monte Carlo settled early as a vehicle for analysis, learning and communication. Other major tools for risk studies are stochastic modelling of dependence (Chapters 5 and 6) and historical estimation and error (Chapter 7). Part I carries mathematical, statistical and computational methods from one application area to another, with examples from everywhere.

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